Circular stairs are worth a look.

https://www.google.com/search?q=images+circular+stairs&tbm=isch&source=hp&sa=X&ved=2ahUKEwi31-jGl4vlAhVynuAKHXx5DuIQsAR6BAgJEAE&biw=1328&bih=617#spf=1570486909891

Certain dimensions will have to follow code.

https://buildingcodetrainer.com/residential-stair-code/

The critical measurement is height required, first floor surface to second floor surface. I think you could do that stairs in an 8x10 space with 7 treads forward to a full width landing and 7 treads back up the other side to the upper floor. I'm not an expert on stairs but I think it might be an option

Stairs get tricky with height and width requirements, headroom issues and then adequate landing space at the top and bottom. I believe it is very rare to find a house where the cost of moving stairs is worth the benefit. Stairs get expensive really quickly!

https://www.homedit.com/alternating-tread-stairs/

https://www.pinterest.ca/pin/25332816631841700/

New stairs would have to meet current code. If your home is older, your stairs may not comply with current code. If so, you cannot move or replicate them elsewhere.

I'll post photos cause it's uncertain if you will or can move them. Idk how much can fit under your stairs, but it's sure a neat idea!

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This doesn't look as expensive as last one -

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If you build new stairs, here a storage idea, and they "soft close."

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Probably an architectural consult would be good for this. They will be able to lay it out for you. The total height (rise) is from finished floor to finished floor. So 8’ 3” + 10 to 12” for the second floor joists and flooring and the first floor ceiling.

One more - possibly in a kitchen, a couple sections of cans would be nice, colorful.

Churchlands Private Residence · More Info

Post a drawing, with measurements.

We need pictures and a layout of the whole house floor plan to scale. Moving stairs is a huge undertaking BTW

Among other things, you'd need to be sure that the structure of the current staircase isn't part of what supports the upper floor in that area. 10' sounds pretty short for a normal flight of stairs, but you do have the width for a turnaround. Be prepared for quotes in the tens of thousands of dollars...

As everyone else has said, moving stairs is difficult and expensive. It might be easier and more price-conscious to just relocate the kitchen somewhere else in the home, but post a floor plan.